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We introduce a two-dimensional Keller-Segel type free boundary model formotility of eukaryotic cells on substrates. The key ingredients of this modelare the Darcy law for overdamped motion of the cytoskeleton (active) gel andHele-Shaw type boundary conditions (Young-Laplace equation for pressure andcontinuity of velocities). We first show that radially symmetric steady statesolutions become unstable and bifurcate to traveling wave solutions. Next weestablish linear and nonlinear stability of the steady states. We show thatlinear stability analysis is inconclusive for both steady states and travelingwaves. Therefore we use invariance properties to prove nonlinear stability ofsteady states.

Stability of steady states and bifurcation to traveling waves in a free boundary model of cell motility